We use mathematical models to study the mechanisms of oscillatory electrical activity arising from ion channels in cell membranes and modulated by intracellular chemical processes. We are interested in both the behavior of single cells and the ways in which cells communicate and modify each other's behavior. Our main application has been to the biophysical basis of insulin secretion in pancreatic beta-cells. We have examined bursting oscillations in membrane potential and the role of electrical coupling between cells in the islet of Langerhans. Long term goals are to understand how the membrane dynamics interact with intracellular events to regulate secretion. We also compare, contrast, and generalize to other secretory cells and neurons, including GnRH-secreting hypothalamic neurons, pituitary somatotrophs, and fast neurotransmitter secretion at nerve terminals. Our primary tool is the numerical solution of ordinary and partial differential equations. We use analytical, geometrical, graphical, and numerical techniques from the mathematical theory of dynamical systems to help construct and interpret the models. Perturbation techniques are used to get analytical results in special cases. We study both detailed biophysical models and simplified models which are more amenable to analysis. Such an approach aids the isolation of the essential or minimal mechanisms underlying phenomena, the search for general principles, and the application of concepts and analogies from other fields. We see a role for our group as intermediaries between the mathematical and biological disciplines. This includes disseminating the insights of mathematical work to biologists in accessible language and alerting mathematicians and other theoreticians to new and challenging problems arising from biological issues. Recent work on this project includes: 1. We developed a model to account for the wide range of oscillation periods for membrane potential and calcium observed in pancreatic beta-cells and islets (from seconds to minutes). The hypothesis was that oscillations are governed by two slow, negative feedback processes, one with a time constant of 1-5 seconds, and one with a time constant of 2 minutes. There is no process with an intermediate time constant - oscillations in that range result from the interaction between the two faster and slower processes. In collaboration with the Satin lab we confirmed a key prediction of the model that appropriate injected currents could elicit medium-scale oscillations from fast cells. This showed for the first time that isolated cells from pancreatic islets could indeed exhibit medium oscillations (Ref. #1). 2. We have studied how electrical coupling of pancreatic beta-cells contributes to medium-scale (10-60 sec) bursting oscillations in calcium and membrane potential observed in pancreatic islets. We focused on the sub-group (30 - 50%) of isolated cells identified by the Satin lab that show rapid and continuous spiking rather than bursts of spikes. We had previously shown that such cells, when electrically coupled, could be transformed into bursters. However, the phenomenon was not robust, existing only for a small range of coupling strengths. We have now shown and analyzed mathematically that nonlinear effects of stochastic ion channel fluctuations (Ref. # 2) and heterogeneity of parameters (Ref. # 3) can enhance this form of emergent bursting. This study provides an interesting contrast and complement to our previous demonstrations that noise and heterogeneity hinder bursting when added to cells that are intrinsic bursters when uncoupled. 3. We have extended our previous studies of store-operated calcium channels (SOC) in pancreatic beta-cells to neuro-endocrine cells of the hypothalamus that secrete GnRH. In collaboration with the Stojilkovic lab, we showed that SOC could account for the increased action potential frequency and increased cytosolic calcium levels seen when the cells are stimulated with GnRH (the cells have autoreceptors for their own secretion product). See Refs. # 4 and 6. 4. We have carried out a detailed mathematical analysis of the steady-state spatial profile of calcium near an open calcium channel and how it is modified by calcium buffers. This resulted in a systematic and unified treatment of several approximate formulas previously obtained by others using heuristic arguments. Such a treatment also indicates in which parameter regimes the various approximations are valid and leads to refined approximations that are more accurate, provided they are used in the appropriate parameter regimes. See Ref. #5.